A Hybrid Modelling of a Water and Air Injector in a Subsonic Icing Wind Tunnel ††thanks: This work was supported by the Dispositif Recherche of Métropole Rouen Normandie through the COPOGIRT project. For this reason and the purpose of Open Access, the aut (2024)

The mass balance law provides the relation

where $m$ is the water mass inside the tank, and $\dot{m}^{\text{w}}_{i}$ and $\dot{m}^{\text{w}}_{o}$ are the input and output fluids mass flow rate inside the tank.

Remark 1.

In the following, the input mass flow rate $\dot{m}^{\text{w}}_{i}$ is a control variable that is chosen by the user depending on the operational needs.In addition, $\dot{m}^{\text{w}}_{o}$ is the water mass flow rate injected into the test section.

In the tank, the overall mass can be divided into a mass of water $m_{\text{w}}$ and a mass of air $m_{\text{wa}}$, so that

However, given the tank’s volume, we have $m_{\text{w}}\gg m_{\text{wa}}$.It follows that the mass of the air can be considered constant, so that

As described before, the tank is cylindrical in shape and $m_{\text{w}}=\rho_{\text{w}}S_{1}h$, with $\rho_{\text{w}}$ the water density in $\mathrm{kg}\text{${\cdot}$}{\mathrm{m}}^{-3}$, so that

Therefore

where $\rho_{\text{w}}$, in $\mathrm{kg}\text{${\cdot}$}{\mathrm{m}}^{-3}$, is the water density.

In terms of temperature, the energy change in volume control is due to several elements.The first source of change is the power provided by the heating system $Q_{\text{w}}$, in $\mathrm{W}$.The second source are the losses caused by leaks to the outside of system $-\kappa_{\text{w}}T_{\text{w}}$ where $\kappa_{\text{w}}$, in $\mathrm{W}\text{${\cdot}$}{\mathrm{\SIUnitSymbolCelsius}}^{-1}$, is a constant characterising the leaks, and $T_{\text{w}}$, in $\mathrm{\SIUnitSymbolCelsius}$, is the temperature in the water tank.The third source is the input energy provided by the incoming mass $m_{i}C_{e}T^{\text{w}}_{i}$, where $C_{e}$, in $\mathrm{J}\text{${\cdot}$}{\mathrm{kg}}^{-1}\text{${\cdot}$}{\mathrm{K}}^{-1}$, is the water’s mass specific heat, and $T^{\text{w}}_{i}$, in $\mathrm{K}$, is the temperature of the incoming water.Finally, the last source is the output energy provided by the outgoing mass $m_{o}C_{e}T^{\text{K}}_{\text{w}}$, where $T^{\text{K}}_{\text{w}}=T_{\text{w}}+273.15$ is the temperature in the water tank expressed in Kelvin.Then, considering the energy $E$, in $\mathrm{J}$, of liquid water at a certain temperature, we have

The energy rate of change is then

and, using (1) into (7),

Given what we have described above, we have

so that

or

According to the technical report [19] of the water valve provided by the manufacturer, the volumetric flow of water through a valve $Q_{\text{wv}}$ is a function of the valve’s opening area $A$, in ${\mathrm{m}}^{2}$, the discharge coefficient $C_{d}$, the pressure difference $\Delta P_{\text{wv}}$, in $\mathrm{Pa}$, in the valve, and the density of water $\rho_{\text{w}}$, in $\mathrm{kg}\text{${\cdot}$}{\mathrm{m}}^{-3}$, and is given by

The discharge coefficient $C_{d}$ is obtained as

where $K_{v}$ is the flow factor, in ${\mathrm{m}}^{3}\text{${\cdot}$}{\mathrm{h}}^{-1}$, given by the manufacturer, and $D$, in $\mathrm{m}$, is the diameter of the valve’s orifice.The difference of pressure $\Delta P_{\text{wv}}$ is obtained as

where $\xi$ is the pressure drop coefficient and $v_{\text{wv}}$, in $\mathrm{m}\text{${\cdot}$}{\mathrm{s}}^{-1}$, is the velocity at which water flows through the valve.The pressure drop coefficient $\xi$ is given by

Through the air valve, the volumetric flow $Q_{\text{av}}$ is given by [20, 21]

where $\gamma=1.4$ is the ratio of specific heats, $R=$2870\text{\,}\mathrm{J}\text{${\cdot}$}{\mathrm{kg}}^{-1}\text{${\cdot}$}{$ is the air-gas constant, $P_{\text{a}}$, in $\mathrm{Pa}$, is the air pressure in the air tank pressure, and $P_{\text{av}}$, in $\mathrm{Pa}$, is the air pressure outside the control flow valve.

With regard to velocity, a very simplified model using Bernoulli’s continuity equation is used to determine the velocity provided by the nozzles towards the test section.Bernoulli’s equation can be viewed as a conservation of energy law for a flowing fluid.Figure 4 presents a schematic view the nozzles’ geometry.The input surface $S_{i}$ is taken $1.2$ times bigger than the outlet surface $S_{o}$.Hence, the velocity at the outlet of the nozzles is given by

where $S_{i}$ and $S_{o}$, in ${\mathrm{m}}^{2}$, are respectively the inlet and outlet surfaces, and $v^{\text{n}}_{i}$ and $v^{\text{n}}_{o}$, in $\mathrm{m}\text{${\cdot}$}{\mathrm{s}}^{-1}$, are the velocity of the air coming from the air tank and of the mix of air and water injected in the test section.It follows that

Remark 2.

As usual for this type of equipment [11], the velocity of the water injected inside the nozzle is not taken into account, as it is negligible.It is the pressure provoked by the air velocity that leads to the spraying of a mix of air and water at the output of the nozzle.

A simple form is imposed to determine the polynomial model giving $T_{\text{n}}$ as a function of $T_{\text{w}}$, $T_{\text{TS}}$, and $T_{\text{a}}$.Therefore, minimizing the MSE, we obtain

where all the temperatures are expressed in degree Celsius.The regression tree giving the value of $T_{\text{n}}$ is presented in Figure 6.It has to be read as tests on the variables $T_{\text{w}}$, $T_{\text{TS}}$, and $T_{\text{a}}$, expressed in degree Celsius.Depending on the position of these variables with given thresholds, indicated on the tree’s branches, we obtain a value, in degree Celsius, for $T_{\text{n}}$.

For the neural model of $T_{\text{n}}$, we optimize the properties of the neural network among the following possibilities:

• •

  the number of hidden layers is either $1$, $2$, or $3$;

• •

  each hidden layer can contain between $2$ and $10$ neurons;

• •

  the activation function is either $sigmoid$, $tanh$, or $relu$.

As for the other models, a 5-fold cross-validation is used and the characteristics of the neural network are obtained through minimisation of the MSE.The optimal characteristics of the neural model of $T_{\text{n}}$ are presented in Table IV.

According to [11], LWC $\Lambda$ represents the mixing of the available mass of water $m^{\text{w}}_{\text{TS}}$ within a defined air volume $V^{\text{a}}_{\text{TS}}$ in the test section, given by

The volume of air flowing through the test section can be calculated by $V^{\text{a}}_{\text{TS}}=v_{\text{TS}}A_{\text{TS}}$ where $A_{\text{TS}}$ is the cross-sectional area of the test section.The available mass of water is $m^{\text{w}}_{\text{TS}}=\rho_{\text{w}}Q_{\text{w}}$.It follows that

To obtain the MVD polynomial, more complex forms than for $T_{\text{n}}$ are imposed.First, as $Q_{\text{a}}$ has the highest influence on the MVD $\mathrm{M}$ according to Figure 9, we propose a polynomial model where $Q_{\text{a}}$ is a factor of all terms composed of all possible combinations of 1 to 3 variables among $\Lambda$, $T_{\text{TS}}$, and $v_{\text{TS}}$ to the power $1$.Minimizing the MSE with this form, we obtain